FORM INVARIANCE OF DIFFERENTIAL-EQUATIONS IN GENERAL-RELATIVITY

Einstein equations for several matter sources in Robertson-Walker and Bianchi I type metrics, are shown to reduce to a kind of second-order nonlinear ordinary differential equation y + alpha f(y)y + beta f(y)in tegral f(y)dy + gamma f(y)=0. Also, it appears in the generalized stat istical mechanics for the most interesting value q=-1. The invariant f orm of this equation is imposed and the corresponding nonlocal transfo rmation is obtained. The linearization of that equation for any alpha, beta, and gamma is presented and for the important case f=by(n) + k w ith beta=alpha(2)(n + 1)/(n + 2)(2) its explicit general solution is f ound. Moreover, the form invariance is applied to yield exact solution s of some other differential equations. (C) 1997 American Institute of Physics.